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If {x} is a continuous, real valued random variable defined over the interval (-∞, +∞) and its occurrence is defined by the density function given as: f(x) = 1/sqrt(2π+b) e^(-1/2((x-a)/b)²) where 'a' and 'b' are the statistical attributes of the random variable {x}. The value of the integral ∫_(-∞)^(∞) 1/sqrt(2π+b) e^(-1/2((x-a)/b)²) dx is

1. 1
2. 0.5
3. π
4. π/2

Correct answer: 1

Solution

The integral of a probability density function over its entire range must equal 1, as it represents the total probability of all possible outcomes. In this case, the given function is a normalized Gaussian (normal) distribution, which is defined to have an integral of 1.

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